Whakaoti i te f(x)=x^4-29x^2+100

$$f ( x ) = x ^ { 4 } - 29 x ^ { 2 } + 100$$

Tauwehe

$$\left(x-5\right)\left(x-2\right)\left(x+2\right)\left(x+5\right)$$

Ngā Upane Otinga

Kia tauwehea ai te kīanga, me whakaoti te whārite ina ōrite ki te 0.

$$x^{4}-29x^{2}+100=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau 100, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±100,±50,±25,±20,±10,±5,±4,±2,±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$x=2$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{4}-29x^{2}+100 ki te x-2, kia riro ko x^{3}+2x^{2}-25x-50. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$x^{3}+2x^{2}-25x-50=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -50, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±50,±25,±10,±5,±2,±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$x=-2$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{3}+2x^{2}-25x-50 ki te x+2, kia riro ko x^{2}-25. Kia tauwehea ai te otinga, me whakaoti te whārite ina ōrite ki te 0.

$$x^{2}-25=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 1 mō te a, te 0 mō te b, me te -25 mō te c i te ture pūrua.

$$x=\frac{0±\sqrt{0^{2}-4\times 1\left(-25\right)}}{2}$$

Mahia ngā tātaitai.

$$x=\frac{0±10}{2}$$

Whakaotia te whārite x^{2}-25=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=-5 x=5$$

Me tuhi anō te kīanga whakatauwehe mā ngā pūtake i riro.

$$\left(x-5\right)\left(x-2\right)\left(x+2\right)\left(x+5\right)$$

Aromātai

$$x^{4}-29x^{2}+100$$